pacman::p_load(olsrr, corrplot, ggpubr, sf, spdep, GWmodel, tmap, tidyverse, gtsummary)Hands-on Exercise 10: Calibrating Hedonic Pricing Model for Private Highrise Property with Geographically weighted regression (GWR)
In this exercise, I am applying GWR to develop hedonic pricing models, where structural and locational variables are used to model 2015 resale condo prices.
R Packages used
OLS and performing diagnostics tests
Calibrating geographical weighted family of models
Multivariate data visualisation and analysis
Spatial data handling
- sf
Attribute data handling
- tidyverse, especially readr, ggplot2 and dplyr
Choropleth mapping
- tmap
GW models suit situations when data are not described well by some global model, but where there are spatial regions where a suitably localised calibration provides a better description.
GWmodel includes functions to calibrate: GW summary statistics, GW principal components analysis, GW discriminant analysis and various forms of GW regression; some of which are provided in basic and robust (outlier resistant) forms.
Data
URA Master Plan subzone boundary
mpsz <- st_read(dsn = 'data/geospatial/', layer = 'MP14_SUBZONE_WEB_PL')Resale prices of condominium in 2015
condo_resale <- read_csv('data/aspatial/Condo_resale_2015.csv')Wrangling
URA Master Plan subzone boundary
Since mpsz does not have EPSG information, use st_tranform() to modify the projection of mpsz
mpsz_3414 <- st_transform(mpsz, 3414)
# to verify
st_crs(mpsz)Reveal the extent of mpsz_3414
# Returns bounding of a simple feature or simple feature set
st_bbox(mpsz_3414)Resale prices of condominium in 2015
summary(condo_resale)condo_resale_sf <- condo_resale %>% st_as_sf(coords = c('LONGITUDE','LATITUDE'), crs=4326) %>%
st_transform(crs=3414) condo_resale_sfEDA using ggplot2
Univariate
SELLING_PRICE
Code
ggplot(data=condo_resale_sf, aes(x=`SELLING_PRICE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
Since it has a right-skewed distribution, normalise using logarithmic transformation.
LOG_SELLING_PRICE
condo_resale_sf <- condo_resale_sf %>%
mutate(`LOG_SELLING_PRICE` = log(SELLING_PRICE))SELLING_PRICE vs LOG_SELLING_PRICE
Code
ggplot(data=condo_resale_sf, aes(x=`SELLING_PRICE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
ggplot(data=condo_resale_sf, aes(x=`LOG_SELLING_PRICE`)) +
geom_histogram(bins=20, color="black", fill="light blue")

Multivariate (Trellis)
Code
AREA_SQM <- ggplot(data=condo_resale_sf, aes(x= `AREA_SQM`)) +
geom_histogram(bins=20, color="black", fill="light blue")
AGE <- ggplot(data=condo_resale_sf, aes(x= `AGE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_CBD <- ggplot(data=condo_resale_sf, aes(x= `PROX_CBD`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_CHILDCARE <- ggplot(data=condo_resale_sf, aes(x= `PROX_CHILDCARE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_ELDERLYCARE <- ggplot(data=condo_resale_sf, aes(x= `PROX_ELDERLYCARE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale_sf,
aes(x= `PROX_URA_GROWTH_AREA`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_HAWKER_MARKET <- ggplot(data=condo_resale_sf, aes(x= `PROX_HAWKER_MARKET`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_KINDERGARTEN <- ggplot(data=condo_resale_sf, aes(x= `PROX_KINDERGARTEN`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_MRT <- ggplot(data=condo_resale_sf, aes(x= `PROX_MRT`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_PARK <- ggplot(data=condo_resale_sf, aes(x= `PROX_PARK`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_PRIMARY_SCH <- ggplot(data=condo_resale_sf, aes(x= `PROX_PRIMARY_SCH`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale_sf,
aes(x= `PROX_TOP_PRIMARY_SCH`)) +
geom_histogram(bins=20, color="black", fill="light blue")
ggarrange(AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE,
PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT,
PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH,
ncol = 3, nrow = 4)
Statistical Point Map
tmap_mode("view")Code
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz_3414)+
tm_polygons() +
tm_shape(condo_resale_sf) +
tm_dots(col = "SELLING_PRICE",
alpha = 0.6,
style="quantile") +
tm_view(set.zoom.limits = c(11,14))tmap_mode("plot")Hedonic Pricing Models in R using lm()
Simple Linear Regression
Obtain and print a summary analysis
Code
condo_slr <- lm(
SELLING_PRICE ~ AREA_SQM,
condo_resale_sf)
summary(condo_slr)
Call:
lm(formula = SELLING_PRICE ~ AREA_SQM, data = condo_resale_sf)
Residuals:
Min 1Q Median 3Q Max
-3695815 -391764 -87517 258900 13503875
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -258121.1 63517.2 -4.064 5.09e-05 ***
AREA_SQM 14719.0 428.1 34.381 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 942700 on 1434 degrees of freedom
Multiple R-squared: 0.4518, Adjusted R-squared: 0.4515
F-statistic: 1182 on 1 and 1434 DF, p-value: < 2.2e-16
Visualise best fit curve on a scatterplot
Code
# using lm() as a method function
ggplot(data=condo_resale_sf,
aes(x=`AREA_SQM`, y=`SELLING_PRICE`)) +
geom_point() +
geom_smooth(method = lm)
Multiple Linear Regression
Visualising the relationships of the independent variables (to identify highly correlated variables)
Code
# corrplot
corrplot(cor(
condo_resale[,5:23]),
diag = FALSE, # correlation coefficients are hidden
order = 'AOE',
tl.pos = 'td', # top-diagonal text labels
tl.cex = .5, # size of text labels
method = 'number', # visualisation method
type = 'upper' # displays upper half triangle of correlation matrix
)
Building the model
Code
# using lm()
condo_mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE +
PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN +
PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH +
PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data=condo_resale_sf)
summary(condo_mlr)
Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE +
PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET +
PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH +
PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data = condo_resale_sf)
Residuals:
Min 1Q Median 3Q Max
-3475964 -293923 -23069 241043 12260381
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 481728.40 121441.01 3.967 7.65e-05 ***
AREA_SQM 12708.32 369.59 34.385 < 2e-16 ***
AGE -24440.82 2763.16 -8.845 < 2e-16 ***
PROX_CBD -78669.78 6768.97 -11.622 < 2e-16 ***
PROX_CHILDCARE -351617.91 109467.25 -3.212 0.00135 **
PROX_ELDERLYCARE 171029.42 42110.51 4.061 5.14e-05 ***
PROX_URA_GROWTH_AREA 38474.53 12523.57 3.072 0.00217 **
PROX_HAWKER_MARKET 23746.10 29299.76 0.810 0.41782
PROX_KINDERGARTEN 147468.99 82668.87 1.784 0.07466 .
PROX_MRT -314599.68 57947.44 -5.429 6.66e-08 ***
PROX_PARK 563280.50 66551.68 8.464 < 2e-16 ***
PROX_PRIMARY_SCH 180186.08 65237.95 2.762 0.00582 **
PROX_TOP_PRIMARY_SCH 2280.04 20410.43 0.112 0.91107
PROX_SHOPPING_MALL -206604.06 42840.60 -4.823 1.57e-06 ***
PROX_SUPERMARKET -44991.80 77082.64 -0.584 0.55953
PROX_BUS_STOP 683121.35 138353.28 4.938 8.85e-07 ***
NO_Of_UNITS -231.18 89.03 -2.597 0.00951 **
FAMILY_FRIENDLY 140340.77 47020.55 2.985 0.00289 **
FREEHOLD 359913.01 49220.22 7.312 4.38e-13 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 755800 on 1417 degrees of freedom
Multiple R-squared: 0.6518, Adjusted R-squared: 0.6474
F-statistic: 147.4 on 18 and 1417 DF, p-value: < 2.2e-16
Revising the Model
[Remove] statistically insignificant variables
condo_mlr1 <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE +
PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK +
PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data=condo_resale_sf)[VIF] ols_vif_tol() - check multicolinearity
Code
ols_vif_tol(condo_mlr1) Variables Tolerance VIF
1 AREA_SQM 0.8728554 1.145665
2 AGE 0.7071275 1.414172
3 PROX_CBD 0.6356147 1.573280
4 PROX_CHILDCARE 0.3066019 3.261559
5 PROX_ELDERLYCARE 0.6598479 1.515501
6 PROX_URA_GROWTH_AREA 0.7510311 1.331503
7 PROX_MRT 0.5236090 1.909822
8 PROX_PARK 0.8279261 1.207837
9 PROX_PRIMARY_SCH 0.4524628 2.210126
10 PROX_SHOPPING_MALL 0.6738795 1.483945
11 PROX_BUS_STOP 0.3514118 2.845664
12 NO_Of_UNITS 0.6901036 1.449058
13 FAMILY_FRIENDLY 0.7244157 1.380423
14 FREEHOLD 0.6931163 1.442759
[Residual] ols_plot_resid_fit() - check linearity
Code
ols_plot_resid_fit(condo_mlr1)
[Residual] ols_plot_resid_hist() or
{#| output: true} #| code-fold: true{r} ols_plot_resid_hist(condo_mlr1)
[Stat. test] ols_test_normality() - check normality
Code
ols_test_normality(condo_mlr1)-----------------------------------------------
Test Statistic pvalue
-----------------------------------------------
Shapiro-Wilk 0.6856 0.0000
Kolmogorov-Smirnov 0.1366 0.0000
Cramer-von Mises 121.0768 0.0000
Anderson-Darling 67.9551 0.0000
-----------------------------------------------
Spatial autocorrelation test
Code
# save residual of the hedonic pricing model as a data frame
mlr_output <- as.data.frame(condo_mlr1$residuals)
# join with condo_resale_sf
condo_resale_res_sf <- cbind(condo_resale_sf, mlr_output) %>%
mutate(MLR_RES = condo_mlr1$residuals)
# convert condo_resale_res_sf into a SpatialPointsDataFrame
condo_resale_sp <- as_Spatial(condo_resale_res_sf)
condo_resale_spclass : SpatialPointsDataFrame
features : 1436
extent : 14940.85, 43352.45, 24765.67, 48382.81 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables : 24
names : POSTCODE, SELLING_PRICE, AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT, PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ...
min values : 18965, 540000, 34, 0, 0.386916393, 0.004927023, 0.054508623, 0.214539508, 0.051817113, 0.004927023, 0.052779424, 0.029064164, 0.077106132, 0.077106132, 0, ...
max values : 828833, 1.8e+07, 619, 37, 19.18042832, 3.46572633, 3.949157205, 9.15540001, 5.374348075, 2.229045366, 3.48037319, 2.16104919, 3.928989144, 6.748192062, 3.477433767, ...
tmap_mode("view")Code
tm_shape(mpsz_3414) +
tmap_options(check.and.fix = TRUE) +
tm_polygons(alpha = 0.4) +
tm_shape(condo_resale_res_sf) +
tm_dots(col = "MLR_RES",
alpha = 0.6,
style="quantile") +
tm_view(set.zoom.limits = c(11,14))tmap_mode('plot')Code
# Moran's I
# compute the distance-based weight matrix
nb <- dnearneigh(
coordinates(condo_resale_sp),
0, # lower bound
1500, # upper bound
longlat = FALSE
)
# convert nb into spatial weights
nb_lw <- nb2listw(nb, style = 'W')
# perform Moran's I
lm.morantest(condo_mlr1, nb_lw)
Global Moran I for regression residuals
data:
model: lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT +
PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data = condo_resale_sf)
weights: nb_lw
Moran I statistic standard deviate = 24.366, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Observed Moran I Expectation Variance
1.438876e-01 -5.487594e-03 3.758259e-05
Code
# summary(nb)
# summary(nb_lw)Generate publication-ready summary tables
Code
ols_regress(condo_mlr1) Model Summary
-----------------------------------------------------------------------------
R 0.807 RMSE 751998.679
R-Squared 0.651 MSE 571471422208.592
Adj. R-Squared 0.647 Coef. Var 43.168
Pred R-Squared 0.638 AIC 42966.758
MAE 414819.628 SBC 43051.072
-----------------------------------------------------------------------------
RMSE: Root Mean Square Error
MSE: Mean Square Error
MAE: Mean Absolute Error
AIC: Akaike Information Criteria
SBC: Schwarz Bayesian Criteria
ANOVA
--------------------------------------------------------------------------------
Sum of
Squares DF Mean Square F Sig.
--------------------------------------------------------------------------------
Regression 1.512586e+15 14 1.080418e+14 189.059 0.0000
Residual 8.120609e+14 1421 571471422208.592
Total 2.324647e+15 1435
--------------------------------------------------------------------------------
Parameter Estimates
-----------------------------------------------------------------------------------------------------------------
model Beta Std. Error Std. Beta t Sig lower upper
-----------------------------------------------------------------------------------------------------------------
(Intercept) 527633.222 108183.223 4.877 0.000 315417.244 739849.200
AREA_SQM 12777.523 367.479 0.584 34.771 0.000 12056.663 13498.382
AGE -24687.739 2754.845 -0.167 -8.962 0.000 -30091.739 -19283.740
PROX_CBD -77131.323 5763.125 -0.263 -13.384 0.000 -88436.469 -65826.176
PROX_CHILDCARE -318472.751 107959.512 -0.084 -2.950 0.003 -530249.889 -106695.613
PROX_ELDERLYCARE 185575.623 39901.864 0.090 4.651 0.000 107302.737 263848.510
PROX_URA_GROWTH_AREA 39163.254 11754.829 0.060 3.332 0.001 16104.571 62221.936
PROX_MRT -294745.107 56916.367 -0.112 -5.179 0.000 -406394.234 -183095.980
PROX_PARK 570504.807 65507.029 0.150 8.709 0.000 442003.938 699005.677
PROX_PRIMARY_SCH 159856.136 60234.599 0.062 2.654 0.008 41697.849 278014.424
PROX_SHOPPING_MALL -220947.251 36561.832 -0.115 -6.043 0.000 -292668.213 -149226.288
PROX_BUS_STOP 682482.221 134513.243 0.134 5.074 0.000 418616.359 946348.082
NO_Of_UNITS -245.480 87.947 -0.053 -2.791 0.005 -418.000 -72.961
FAMILY_FRIENDLY 146307.576 46893.021 0.057 3.120 0.002 54320.593 238294.560
FREEHOLD 350599.812 48506.485 0.136 7.228 0.000 255447.802 445751.821
-----------------------------------------------------------------------------------------------------------------
OR,
Code
tbl_regression(condo_mlr1, intercept = TRUE) %>%
# include model statistics as a table source note
add_glance_source_note(
label = list(sigma ~ "\U03C3"),
include = c(
r.squared,
adj.r.squared,
AIC,
statistic,
p.value,
sigma))| Characteristic | Beta | 95% CI1 | p-value |
|---|---|---|---|
| (Intercept) | 527,633 | 315,417, 739,849 | <0.001 |
| AREA_SQM | 12,778 | 12,057, 13,498 | <0.001 |
| AGE | -24,688 | -30,092, -19,284 | <0.001 |
| PROX_CBD | -77,131 | -88,436, -65,826 | <0.001 |
| PROX_CHILDCARE | -318,473 | -530,250, -106,696 | 0.003 |
| PROX_ELDERLYCARE | 185,576 | 107,303, 263,849 | <0.001 |
| PROX_URA_GROWTH_AREA | 39,163 | 16,105, 62,222 | <0.001 |
| PROX_MRT | -294,745 | -406,394, -183,096 | <0.001 |
| PROX_PARK | 570,505 | 442,004, 699,006 | <0.001 |
| PROX_PRIMARY_SCH | 159,856 | 41,698, 278,014 | 0.008 |
| PROX_SHOPPING_MALL | -220,947 | -292,668, -149,226 | <0.001 |
| PROX_BUS_STOP | 682,482 | 418,616, 946,348 | <0.001 |
| NO_Of_UNITS | -245 | -418, -73 | 0.005 |
| FAMILY_FRIENDLY | 146,308 | 54,321, 238,295 | 0.002 |
| FREEHOLD | 350,600 | 255,448, 445,752 | <0.001 |
| R² = 0.651; Adjusted R² = 0.647; AIC = 42,967; Statistic = 189; p-value = <0.001; σ = 755,957 | |||
| 1 CI = Confidence Interval | |||
Code
# More info
# https://www.danieldsjoberg.com/gtsummary/articles/tbl_regression.htmlRDS
write_rds(mpsz_3414,'../ex11/data/rds/mpsz_3414.rds')
write_rds(condo_resale_sp,'../ex11/data/rds/condo_resale_sp.rds')